Applications of the Ashtekar gravity to four dimensional hyperk\"ahler geometry and Yang-Mills Instantons

TL;DR

This paper explores how Ashtekar gravity equations can be applied to construct hyperk"ahler metrics and anti self-dual Yang-Mills connections on four-dimensional manifolds, linking geometric structures with gauge theory solutions.

Contribution

It introduces a method to generate hyperk"ahler metrics and anti self-dual Yang-Mills connections using Ashtekar equations and reductions of anti self-dual connections, expanding the toolkit for geometric and gauge-theoretic constructions.

Findings

Explicit hyperk"ahler metrics constructed from Ashtekar equations

Anti self-dual Yang-Mills connections derived via reductions

Solutions obtained through Nahm and Laplace equations

Abstract

The Ashtekar-Mason-Newman equations are used to construct the hyperk\"ahler metrics on four dimensional manifolds. These equations are closely related to anti self-dual Yang-Mills equations of the infinite dimensional gauge Lie algebras of all volume preserving vector fields. Several examples of hyperk\"ahler metrics are presented through the reductions of anti self-dual connections. For any gauge group anti self-dual connections on hyperk\"ahler manifolds are constructed using the solutions of both Nahm and Laplace equations.